2,688 research outputs found
On the volume of the convex hull of two convex bodies
In this note we examine the volume of the convex hull of two congruent copies
of a convex body in Euclidean -space, under some subsets of the isometry
group of the space. We prove inequalities for this volume if the two bodies are
translates, or reflected copies of each other about a common point or a
hyperplane containing it. In particular, we give a proof of a related
conjecture of Rogers and Shephard.Comment: 9 pages, 3 figure
Tube formulas and complex dimensions of self-similar tilings
We use the self-similar tilings constructed by the second author in
"Canonical self-affine tilings by iterated function systems" to define a
generating function for the geometry of a self-similar set in Euclidean space.
This tubular zeta function encodes scaling and curvature properties related to
the complement of the fractal set, and the associated system of mappings. This
allows one to obtain the complex dimensions of the self-similar tiling as the
poles of the tubular zeta function and hence develop a tube formula for
self-similar tilings in \. The resulting power series in
is a fractal extension of Steiner's classical tube formula for
convex bodies K \ci \bRd. Our sum has coefficients related to the curvatures
of the tiling, and contains terms for each integer , just as
Steiner's does. However, our formula also contains terms for each complex
dimension. This provides further justification for the term "complex
dimension". It also extends several aspects of the theory of fractal strings to
higher dimensions and sheds new light on the tube formula for fractals strings
obtained in "Fractal Geometry and Complex Dimensions" by the first author and
Machiel van Frankenhuijsen.Comment: 41 pages, 6 figures, incorporates referee comments and references to
new result
2-perfect m-cycle systems
AbstractThe spectrum for 2-perfect m-cycle systems of Kn has been considered by several authors in the case when m⩜7. In this paper we essentially solve the problem for 2-perfect m-cycle systems of Kn in the case where m is prime and 2m+1 is a prime power. In particular we settle the problem for m = 11 and 13 except for two or one possible exceptions respectively. The problem for m = 9 is also considered
Periodicity of certain piecewise affine planar maps
We determine periodic and aperiodic points of certain piecewise affine maps
in the Euclidean plane. Using these maps, we prove for
that all integer
sequences satisfying are periodic
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